This is a transform which tightens the problem and asks the reader to consult the bracelet literature.

Consider first the question as specified with the additional proviso that the number of beads is fixed at k.

The number of such bracelets is the same as the number of bracelets with k black and n white beads, if 0 is a color on the original bracelet, otherwise k black and n-k white beads. I assume 0 is not a color and let someone else deal with the case that 0 is a color. I also assume k is at most n and at least 1.

Since many of the black and white bracelets do not map to themselves under rotation and reflection, we get a lower bound of (n choose k)/2k. For more precise values, one needs to look at cyclic bracelets with a period p dividing gcd(k,n) as well as for each p considering those invariant under reflection. This analysis should be part of the bracelet literature.

Finally, as the problem did not fix k, one needs presumably to sum over all considered k to get the answer, which should be greater than the number of partitions of n.

Gerhard "Email Me About System Design" Paseman, 2011.06.23